Natlight.mle2: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natlight.bend as Natlight
Lines 26-29 of bend-collections-laws-math@1.0.0.0/proofs/math/typed/natlight.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natlight.bend&start_line=26&end_line=29

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend as NF

def mle2(+a: Nat, +b: Nat, +c: Nat, +d: Nat, +hab: {Nat.is_le(a, b) == True{} : Bool}, +hcd: {Nat.is_le(c, d) == True{} : Bool}) -> {Nat.is_le(Nat.mul(a, c), Nat.mul(b, d)) == True{} : Bool}:
  +l1 = NF.mul_le_r(a, c, d, hcd)
  +l2 = L.subst(Nat, z => {Nat.is_le(z, Nat.mul(d, b)) == True{} : Bool}, Nat.mul(d, a), Nat.mul(a, d), NA.mul_comm(d, a), NF.mul_le_r(d, a, b, hab))
  N.le_trans(Nat.mul(a, c), Nat.mul(a, d), Nat.mul(b, d), l1, L.subst(Nat, z => {Nat.is_le(Nat.mul(a, d), z) == True{} : Bool}, Nat.mul(d, b), Nat.mul(b, d), NA.mul_comm(d, b), l2))

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_comm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/natfuel.bend&name=mul_le_r